Noise reduced circuits for superconducting quantum computers

ABSTRACT

Embodiments described herein are generally related to a method and a system for performing a computation using a hybrid quantum-classical computing system, and, more specifically, to providing an approximate solution to an optimization problem using a hybrid quantum-classical computing system that includes a group of trapped ions. A hybrid quantum-classical computing system that is able to provide a solution to a combinatorial optimization problem may include a classical computer, a system controller, and a quantum processor. The methods and systems described herein include an efficient and noise resilient method for constructing trial states in the quantum processor in solving a problem in a hybrid quantum-classical computing system, which provides improvement over the conventional method for computation in a hybrid quantum-classical computing system.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit to U.S. Provisional Application No.62/852,269, filed May 23, 2019, which is incorporated by referenceherein.

BACKGROUND Field

The present disclosure generally relates to a method of performingcomputation in a hybrid quantum-classical computing system, and morespecifically, to a method of solving an optimization problem in a hybridcomputing system that includes a classical computer and quantum computerthat includes a series of Josephson junctions.

Description of the Related Art

In quantum computing, quantum bits or qubits, which are analogous tobits representing a “0” and a “1” in a classical (digital) computer, arerequired to be prepared, manipulated, and measured (read-out) with nearperfect control during a computation process. Imperfect control of thequbits leads to errors that can accumulate over the computation process,limiting the size of a quantum computer that can perform reliablecomputations.

Among physical systems upon which it is proposed to build large-scalequantum computers is a series of Josephson junctions, each of which isformed by separating two superconducting electrodes with an insulatinglayer that is thin enough to allow pairs of electrons (Cooper pairs) totunnel between the superconducting electrodes. Thus, current (referredto as Josephson current or persistent current) flows between thesuperconducting electrodes in the absence of a bias voltage appliedbetween. A Josephson junction may be modelled as a non-linear resonatorformed from a non-linear current-dependent inductance L_(J) (I) inparallel with a capacitance C). The capacitance C_(J) is determined by aratio of the area of the Josephson junction to the thickness of theinsulating layer. The inductance L_(J) (I) is determined by the Josephcurrent through the insulating layer. Thus, two lowest energy states ofthe non-linear resonator can be used as computational states of a qubit(referred to as “qubit states”). These states can be controlled viamicrowave irradiation of the superconducting circuit.

In current state-of-the-art quantum computers, control of qubits isimperfect (noisy) and the number of qubits used in these quantumcomputers generally range from a hundred qubits to thousands of qubits.The number of quantum gates that can be used in such a quantum computer(referred to as a “noisy intermediate-scale quantum device” or “NISQdevice”) to construct circuits to run an algorithm within a controllederror rate is limited due to the noise.

For solving some optimization problems, a NISQ device having shallowcircuits (with small number of gate operations to be executed intime-sequence) can be used in combination with a classical computer(referred to as a hybrid quantum-classical computing system). Inparticular, finding low-energy states of a many-particle quantum system,such as large molecules, or in finding an approximate solution tocombinatorial optimization problems, a quantum subroutine, which is runon a NISQ device, can be run as part of a classical optimizationroutine, which is run on a classical computer. The classical computer(also referred to as a “classical optimizer”) instructs a controller toprepare the NISQ device (also referred to as a “quantum processor”) inan N-qubit state, execute quantum gate operations, and measure anoutcome of the quantum processor. Subsequently, the classical optimizerinstructs the controller to prepare the quantum processor in a slightlydifferent N-qubit state, and repeats execution of the gate operation andmeasurement of the outcome. This cycle is repeated until the approximatesolution can be extracted. Such hybrid quantum-classical computingsystem having an NISQ device may outperform classical computers infinding low-energy states of a many-particle quantum system and infinding approximate solutions to such combinatorial optimizationproblems. However, the number of quantum gate operations required withinthe quantum routine increases rapidly as the problem size increases,leading to accumulated errors in the NISQ device and causing theoutcomes of these processes to be not reliable.

Therefore, there is a need for a procedure to construct shallow circuitsthat require a minimum number of quantum gate operations to performcomputation and thus reduce noise in a hybrid quantum-classicalcomputing system.

SUMMARY

A method of performing computation in a hybrid quantum-classicalcomputing system includes computing, by a classical computer, a modelHamiltonian including a plurality of sub-Hamiltonian onto which aselected problem is mapped, setting a quantum processor in an initialstate, where the quantum processor comprises a plurality of trappedions, each of which has two frequency-separated states defining a qubit,transforming the quantum processor from the initial state to a trialstate based on each of the plurality of sub-Hamiltonians and an initialset of variational parameters by applying a reduced trial statepreparation circuit to the quantum processor, measuring an expectationvalue of each of the plurality of sub-Hamiltonians on the quantumprocessor, and determining, by the classical computer, if a differencebetween the measured expectation value of the model Hamiltonian is moreor less than a predetermined value. If it is determined that thedifference is more than the predetermined value, the classical computereither selects another set of variational parameters based on aclassical optimization method, sets the quantum processor in the initialstate, transforms the quantum processor from the initial state to a newtrial state based on each of the plurality of sub-Hamiltonians and theanother set of variational parameters by applying a new reduced trialstate preparation circuit to the quantum processor, and measures anexpectation value of the each of the plurality of sub-Hamiltonians onthe quantum processor after transforming the quantum processor to thenew trial state. If it is determined that the difference is less thanthe predetermined value, the classical computer outputs the measuredexpectation value of the model Hamiltonian as an optimized solution tothe selected problem.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above-recited features of the presentdisclosure can be understood in detail, a more particular description ofthe disclosure, briefly summarized above, may be had by reference toembodiments, some of which are illustrated in the appended drawings. Itis to be noted, however, that the appended drawings illustrate onlytypical embodiments of this disclosure and are therefore not to beconsidered limiting of its scope, for the disclosure may admit to otherequally effective embodiments.

FIG. 1 is a schematic partial view of a superconducting qubit-basedquantum computing system according to one embodiment.

FIG. 2 depicts a schematic view of a schematic view of a quantumprocessor according to one embodiment.

FIG. 3A is a schematic view of a Josephson junction according to oneembodiment.

FIGS. 3B and 3C are schematic models of a Josephson junction accordingto one embodiment.

FIG. 3D is a schematic view of a circuit for forming a charge qubitaccording to one embodiment.

FIG. 3E is a schematic view of a circuit for forming a flux qubitaccording to one embodiment.

FIG. 4 depicts a qubit state of a superconducting qubit represented as apoint on a surface of the Bloch sphere.

FIG. 5 depicts an overall hybrid quantum-classical computing system forobtaining a solution to an optimization problem by Variational QuantumEigensolver (VQE) algorithm or Quantum Approximate OptimizationAlgorithm (QAOA) according to one embodiment.

FIG. 6 depicts a flowchart illustrating a method of obtaining a solutionto an optimization problem by Variational Quantum Eigensolver (VQE)algorithm or Quantum Approximate Optimization Algorithm (QAOA) accordingto one embodiment.

FIG. 7A illustrates a trial state preparation circuit according to oneembodiment.

FIG. 7B illustrates reduced trial state preparation circuits accordingto one embodiment.

To facilitate understanding, identical reference numerals have beenused, where possible, to designate identical elements that are common tothe figures. In the figures and the following description, an orthogonalcoordinate system including an X-axis, a Y-axis, and a Z-axis is used.The directions represented by the arrows in the drawing are assumed tobe positive directions for convenience. It is contemplated that elementsdisclosed in some embodiments may be beneficially utilized on otherimplementations without specific recitation.

DETAILED DESCRIPTION

Embodiments described herein are generally related to a method and asystem for performing a computation using a hybrid quantum-classicalcomputing system, and, more specifically, to providing an approximatesolution to an optimization problem using a hybrid quantum-classicalcomputing system that includes a series of Josephson junctions.

A hybrid quantum-classical computing system that is able to provide asolution to a combinatorial optimization problem may include a classicalcomputer, a system controller, and a quantum processor. In someembodiments, the system controller is housed within the classicalcomputer. The classical computer performs supporting and system controltasks including selecting a combinatorial optimization problem to be runby use of a user interface, running a classical optimization routine,translating the series of logic gates into pulses to apply on thequantum processor, and pre-calculating parameters that optimize thepulses by use of a central processing unit (CPU). A software program forperforming the tasks is stored in a non-volatile memory within theclassical computer.

The quantum processor includes a series of Josephson junctions that arecoupled with various hardware, including microwave generators,synthesizers, mixers, and resonators to manipulate and to read-out thestates of superconducting qubits. The system controller receives fromthe classical computer instructions for controlling the quantumprocessor, controls various hardware associated with controlling any andall aspects used to run the instructions for controlling the quantumprocessor, and returns a read-out of the quantum processor and thusoutput of results of the computation(s) to the classical computer.

The methods and systems described herein include an efficient method forconstructing quantum gate operations executed by the quantum processorin solving a problem in a hybrid quantum-classical computing system.

General Hardware Configurations

FIG. 1 is a schematic partial view of a superconducting qubit-basedquantum computing system, or system 100, according to one embodiment.The system 100 includes a classical (digital) computer 102, a systemcontroller 104 and a quantum processor 106 located at a bottom of adilution refrigerator 108. The classical computer 102 includes a centralprocessing unit (CPU), memory, and support circuits (or I/O). The memoryis connected to the CPU, and may be one or more of a readily availablememory, such as a read-only memory (ROM), a random access memory (RAM),floppy disk, hard disk, or any other form of digital storage, local orremote. Software instructions, algorithms and data can be coded andstored within the memory for instructing the CPU. The support circuits(not shown) are also connected to the CPU for supporting the processorin a conventional manner. The support circuits may include conventionalcache, power supplies, clock circuits, input/output circuitry,subsystems, and the like.

The system controller 104 controls a microwave waveform generator 110that includes synthesizers, waveform generators, and mixers to generatemicrowave pulses. These microwave pulses are filtered by a filter 112and attenuated by an attenuator 114, and provided to the quantumprocessor 106 to perform operations within the quantum processor 106.Analog readout signals from the quantum processor 106 are then amplifiedby an amplifier 116 and digitally processed either on classicalcomputers or in customized field programmable gate arrays (FPGAs) 118for fast processing. The system controller 104 includes a centralprocessing unit (CPU) 120, a read-only memory (ROM) 122, a random accessmemory (RAM) 124, a storage unit 126, and the like. The CPU 120 is aprocessor of the system controller 104. The ROM 122 stores variousprograms and the RAM 124 is the working memory for various programs anddata. The storage unit 126 includes a nonvolatile memory, such as a harddisk drive (HDD) or a flash memory, and stores various programs even ifpower is turned off. The CPU 120, the ROM 122, the RAM 124, and thestorage unit 126 are interconnected via a bus 128. The system controller104 executes a control program which is stored in the ROM 122 or thestorage unit 126 and uses the RAM 124 as a working area. The controlprogram will include software applications that include program codethat may be executed by processor in order to perform variousfunctionalities associated with receiving and analyzing data andcontrolling any and all aspects of the methods and hardware used tocreate the superconducting qubit-based quantum computer system 100discussed herein.

FIG. 2 shows a schematic view of the quantum processor 106. The quantumprocessor 106 is a series of Josephson junctions 202, each of which isused to construct a qubit. In FIG. 2 , an arrow indicates the directionof two-qubit gates, in which the arrow points from a control qubit to atarget qubit (i.e., a two-qubit gate operates on the target qubit basedon a state of the control qubit). An arrangement of Josephson junctions202 in a two-dimensional square lattice form is shown as an example inFIG. 2 . However, the arrangement of the Josephson junctions 202 is notlimited to this example. Controls of states of the Josephson junctions202 are performed by irradiation of the Josephson junctions 202 withmicrowave pulses generated by the microwave waveform generator 110 atroom temperature. Noise in these controls is filtered and attenuated bythe filter 112 and the attenuator 114, respectively, disposed within thedilution refrigerator 108. Signal for readout of the qubit statesfollowing a desired operation on the quantum processor 106 is amplifiedby the amplifier 116 disposed within the dilution refrigerator 108 andmeasured by a known homodyne detection technique.

FIG. 3A is a schematic view of each Josephson junction 202 including twosuperconducting electrodes 302 with an insulating layer 304 between thetwo superconducting electrodes 302. The insulating layer 304 is thinenough to allow pairs of electrons (Cooper pairs) to tunnel between thesuperconducting electrodes 302. Thus, current (Josephson current) flowsbetween the superconducting electrodes 302 even in the absence of a biasvoltage applied in-between. A Josephson junction 202 can be modelled asa non-linear resonator formed from a non-linear current-dependentinductor 306 in parallel with a capacitor 308 as shown in FIG. 3B. Thiscombination of the inductor 306 and the capacitor 308 may be drawn as asingle element for simplicity as shown in FIG. 3C. The capacitance C ofthe capacitor 308 is determined by a ratio of the area of the Josephsonjunction 202 to the thickness of the insulating layer 304. Theinductance L (I) of the inductor 306 is determined by the currentthrough the insulating layer 304. As LC circuits are generally known tomathematically correspond to harmonic oscillators, and thus havediscrete energy states, two lowest energy states of the non-linearresonator can be used as computational states of a qubit (referred to as“qubit states”). These states can be controlled via microwave pulsesgenerated by the microwave waveform generator 110.

A Josephson junction 202 (a non-linear resonator) can provide threequbit archetypes, charge qubits, flux qubits, and phase qubits, andtheir variants (e.g., Fluxonium, Transmon, Xmon, Quantronium) dependingon a circuit 310 to which the Josephson junction 202 is connected. Forany qubit archetypes, qubit states are mapped to different states of thenon-linear resonator, for example, discrete energy states of thenon-linear resonator or to their quantum superpositions. For example, ina charge qubit as shown in FIG. 3D, the different energy levelscorrespond to different integer numbers of Cooper pairs on asuperconducting island 312 (encircled with a dashed line) definedbetween a lead 314 of a capacitor 316 and a Josephson junction 202. Thelowest-lying states are used as qubit states. A gate voltage 318capacitively coupled the superconducting island 312 controls an offsetin the number of Cooper pairs in the superconducting island 312. In aflux qubit as shown in FIG. 3F, the different energy levels correspondto different integer number numbers of magnetic flux quanta trapped in asuperconducting loop 320 including an inductor 322 and one or moreJosephson junctions 202. In the superconducting loop 320, persistentcurrent flows clockwisely and counter-clockwisely when an externalmagnetic flux is applied by an inductively coupled superconductingquantum interference device (DC-SQUID) 324 through a coil 362. When theexternal magnetic flux is close to a half integer number of magneticflux quanta, the two lowest-lying states are symmetric andanti-symmetric superposition states of a clockwise persistent currentstate and a counter-clockwise persistent current state and are be usedas qubit states.

FIG. 4 is provided to help visualize a qubit state of a superconductingqubit is represented as a point on a surface of the Bloch sphere 400with an azimuthal angle ϕ and a polar angle θ. Application of thecomposite pulse as described above, causes Rabi flopping between thequbit state |0

(represented as the north pole of the Bloch sphere) and |1

(the south pole of the Bloch sphere) to occur. Adjusting time durationand amplitudes of the composite pulse flips the qubit state from |0

to |1

(i.e., from the north pole to the south pole of the Bloch sphere), orthe qubit state from |1

to |0

(i.e., from the south pole to the north pole of the Bloch sphere). Thisapplication of the composite pulse is referred to as a “π-pulse”.Further, by adjusting time duration and amplitudes of the compositepulse, the qubit state |0

may be transformed to a superposition state |0

+|1

, where the two qubit states |0

and |1

are added and equally-weighted in-phase (a normalization factor of thesuperposition state is omitted hereinafter without loss of generality)and the qubit state |1

to a superposition state |0

−|1

, where the two qubit states |0

and |1

are added equally-weighted but out of phase. This application of thecomposite pulse is referred to as a “π/2-pulse”. More generally, asuperposition of the two qubits states |0

and |1

that are added and equally-weighted is represented by a point that lieson the equator of the Bloch sphere. For example, the superpositionstates |0

±|1

correspond to points on the equator with the azimuthal angle ϕ beingzero and π, respectively. The superposition states that correspond topoints on the equator with the azimuthal angle ϕ are denoted as |0

+e^(iϕ)|1

(e.g., |0

±i|1) for ϕ=±π/2). Transformation between two points on the equator(i.e., a rotation about the Z-axis on the Bloch sphere) can beimplemented by shifting phases of the composite pulse.

Entanglement Formation

Superconducting qubits can be coupled with their neighboringsuperconducting qubits. In the charge qubits, neighboring qubits can becoupled capacitively, either directly or indirectly mediated by aresonator, which generates interaction between the neighboring chargequbits. The strength of the interaction between the neighboring chargequbits can be tuned by changing the strength of the capacitive coupling,and in case of the indirect mediation, a frequency detuning between thequbits and the resonator.

In the flux qubits, neighboring qubits can be coupled inductivelyresulting in interaction between the neighboring flux qubits. Thestrength of the interaction between the neighboring flux qubits can betuned by dynamically tuning frequency between qubit states of the fluxqubits or frequency of some separate sub-circuit, or by using microwavepulses. Example methods known in the art for tuning of the strength ofinteraction between neighboring superconducting qubits include thedirect-resonant iSWAP, the higher-level resonance induced dynamicalc-Phase (DP), the resonator sideband induced iSWAP, the cross-resonance(CR) gate, the Bell-Rabi, the microwave activated phase gate, and thedriven resonator induced c-Phase (RIP).

By controlling the interaction between two neighboring superconductingqubits (i-th and j-th qubits) as described above in combination withsingle qubit gates, a two-qubit entanglement gate operation, such as acontrolled-NOT operation, or a controlled-Z operation, may be performedon the two neighboring superconducting qubits (i-th and j-th qubits).

The entanglement interaction between two qubits described above can beused to perform a two-qubit entanglement gate operation. The two-qubitentanglement gate operation (entangling gate) along with single-qubitoperations (R gates) forms a set of gates that can be used to build aquantum computer that is configured to perform desired computationalprocesses. Here, the R gate corresponds to manipulation of individualstates of superconducting qubits, and the two-qubit entanglement gate(also referred to as an “entangling gate”) corresponds to manipulationof the entanglement of two neighboring superconducting qubits.

Hybrid Quantum-Classical Computing System

While currently available quantum computers may be noisy and prone toerrors, a combination of both quantum and classical computers, in whicha quantum computer is a domain-specific accelerator, may be able tosolve optimization problems that are beyond the reach of classicalcomputers. An example of such optimization problems is quantumchemistry, where Variational Quantum Eigensolver (VQE) algorithmsperform a search for the lowest energy (or an energy closest to thelowest energy) of a many-particle quantum system and the correspondingstate (e.g. a configuration of the interacting electrons or spins) byiterating computations between a quantum processor and a classicalcomputer. A many-particle quantum system in quantum theory is describedby a model Hamiltonian and the energy of the many-particle quantumsystem corresponds to the expectation value of the model Hamiltonian. Insuch algorithms, a configuration of electrons or spins that is bestknown approximation calculated by the classical computer is input to thequantum processor as a trial state and the energy of the trial state isestimated using the quantum processor. The classical computer receivesthis estimate, modifies the trial state by a known classicaloptimization algorithm, and returns the modified trial state back to thequantum processor. This iteration is repeated until the estimatereceived from the quantum processor is within a predetermined accuracy.A trial function (i.e., a possible configuration of electrons or spinsof the many-particle quantum system) would require exponentially largeresource to represent on a classical computer, as the number ofelectrons or spins of the many-particle quantum system of interest, butonly require linearly-increasing resource on a quantum processor. Thus,the quantum processor acts as an accelerator for the energy estimationsub-routine of the computation. By solving for a configuration ofelectrons or spins having the lowest energy under differentconfigurations and constraints, a range of molecular reactions can beexplored as part of the solution to this type of optimization problemfor example.

Another example optimization problem is in solving combinatorialoptimization problems, where Quantum Approximate Optimization Algorithm(QAOA) perform search for optimal solutions from a set of possiblesolutions according to some given criteria, using a quantum computer anda classical computer. The combinatorial optimization problems that canbe solved by the methods described herein may include the PageRank (PR)problem for ranking web pages in search engine results and themaximum-cut (MaxCut) problem with applications in clustering, networkscience, and statistical physics. The MaxCut problem aims at groupingnodes of a graph into two partitions by cutting across links betweenthem in such a way that a weighted sum of intersected edges ismaximized. The combinatorial optimization problems that can be solved bythe methods described herein may further include the travelling salesmanproblem for finding shortest and/or cheapest round trips visiting allgiven cities. The travelling salesman problem is applied to scheduling aprinting press for a periodical with multi-editions, scheduling schoolbuses minimizing the number of routes and total distance while no bus isoverloaded or exceeds a maximum allowed policy, scheduling a crew ofmessengers to pick up deposit from branch banks and return the depositto a central bank, determining an optimal path for each army planner toaccomplish the goals of the mission in minimum possible time, designingglobal navigation satellite system (GNSS) surveying networks, and thelike. Another combinatorial optimization problem is the knapsack problemto find a way to pack a knapsack to get the maximum total value, givensome items. The knapsack problem is applied to resource allocation givenfinancial constraints in home energy management, network selection formobile nodes, cognitive radio networks, sensor selection in distributedmultiple radar, or the like.

A combinatorial optimization problem is modeled by an objective function(also referred to as a cost function) that maps events or values of oneor more variables onto real numbers representing “cost” associated withthe events or values and seeks to minimize the cost function. In somecases, the combinatorial optimization problem may seek to maximize theobjective function. The combinatorial optimization problem is furthermapped onto a simple physical system described by a model Hamiltonian(corresponding to the sum of kinetic energy and potential energy of allparticles in the system) and the problem seeks the low-lying energystate of the physical system, as in the case of the Variational QuantumEigensolver (VQE) algorithm.

This hybrid quantum-classical computing system has at least thefollowing advantages. First, an initial guess is derived from aclassical computer, and thus the initial guess does not need to beconstructed in a quantum processor that may not be reliable due toinherent and unwanted noise in the system. Second, a quantum processorperforms a small-sized (e.g., between a hundred qubits an a few thousandqubits) but accelerated operation (that can be performed using a smallnumber of quantum logic gates) between an input of a guess from theclassical computer and a measurement of a resulting state, and thus aNISQ device can execute the operation without accumulating errors. Thus,the hybrid quantum-classical computing system may allow challengingproblems to be solved, such as small but challenging combinatorialoptimization problems, which are not practically feasible on classicalcomputers, or suggest ways to speed up the computation with respect tothe results that would be achieved using the best known classicalalgorithm.

FIGS. 5 and 6 depict an overall hybrid quantum-classical computingsystem 500 and a flowchart illustrating a method 600 of obtaining asolution to an optimization problem by Variational Quantum Eigensolver(VQE) algorithm or Quantum Approximate Optimization Algorithm (QAOA)according to one embodiment. In this example, the quantum processor 106is a series of N Josephson junctions 202.

The VQE algorithm relies on a variational search by the well-knownRayleigh-Ritz variational principle. This principle can be used both forsolving quantum chemistry problems by the VQE algorithm andcombinatorial optimization problems solved by the QAOA. The variationalmethod consists of iterations that include choosing a “trial state” ofthe quantum processor depending on a set of one or more parameters(referred to as “variational parameters”) and measuring an expectationvalue of the model Hamiltonian (e.g., energy) of the trial state. A setof variational parameters (and thus a corresponding trial state) isadjusted and an optimal set of variational parameters are found thatminimizes the expectation value of the model Hamiltonian (the energy).The resulting energy is an approximation to the exact lowest energystate. As the processes for obtaining a solution to an optimizationproblem by the VQE algorithm and by the QAOA, the both processes aredescribed in parallel below.

In block 602, by the classical computer 102, an optimization problem tobe solved by the VQE algorithm or the QAOA is selected, for example, byuse of a user interface of the classical computer 102, or retrieved fromthe memory of the classical computer 102, and a model Hamiltonian H_(C),which describes a many-particle quantum system in the quantum chemistryproblem, or to which the selected combinatorial optimization problem ismapped, is computed.

In a quantum chemistry problem defined on an N-spin system, the systemcan be well described by a model Hamiltonian that includes quantum spins(each denoted by the third Pauli matrix σ_(i) ^(z)) (i=1, 2, . . . , N)and couplings among the quantum spins σ_(i) ^(z), H_(C)=Σ_(α=1)^(t)h_(α)P_(α), where P_(α) is a Pauli string (also referred to as aPauli term) P_(α)=σ₁ ^(α1)⊗σ₂ ^(α) ² ⊗ . . . σ_(N) ^(α) ^(N) and σ_(i)^(α) ^(i) is either the identity operator I or the Pauli matrix σ_(i)^(X), σ_(i) ^(Y), or σ_(i) ^(z). Here t stands for the number ofcouplings among the quantum spins and h_(α) (α=1, 2, . . . , t) standsfor the strength of the coupling α. An N-electron system can be alsodescribed by the same model Hamiltonian H_(C)=Σ_(α=1) ^(t)h_(α) P_(α).The goal is to find low-lying energy states of the model HamiltonianH_(C).

In a combinatorial optimization problem defined on a set of N binaryvariables with t constrains (α=1, 2, . . . , t), the objective functionis the number of satisfied clauses C(z)=Σ_(α=1) ^(t)C_(α) (z) or aweighted sum of satisfied clauses C(z)=Σ_(α=1) ^(t)h_(α)C_(α) (z) (h_(α)corresponds to a weight for each constraint α), where z=z₁z₂ . . . z_(N)is a N-bit string and C_(α)(z)=1 if z satisfies the constraint α. Theclause C_(α)(z) that describes the constraint α typically includes asmall number of variables z_(i). The goal is to minimize the objectivefunction. Minimizing this objective function can be converted to findinga low-lying energy state of a model Hamiltonian H_(C)=Σ_(α=1)^(t)h_(α)P_(α)by mapping each binary variable z_(i) to a quantum spinσ_(i) ^(z) and the constraints to the couplings among the quantum spinsσ_(i) ^(z), where P_(α)is a Pauli string (also referred to as a Pauliterm) P_(α)=σ₁ ^(α) ¹ ⊗σ₂ ^(α) ² ⊗ . . . σ_(N) ^(α) ^(N) and α_(i) ^(α)^(i) is either the identity operator I or the Pauli matrix σ_(i) ^(X),σ_(i) ^(Y), or σ_(i) ^(z). Here t stands for the number of couplingsamong the quantum spins and h_(α)(α=1, 2, . . . , t) stands for thestrength of the coupling α.

The quantum processor 106 has N qubits and each quantum spin σ_(i) ^(z)(i=1, 2, . . . , N) is encoded in qubit i (i=1, 2, . . . , N) in thequantum processor 106. For example, the spin-up and spin-down states ofthe quantum spin σ_(i) ^(z) are encoded as |0

and |1

of the qubit i.

In block 604, following the mapping of the selected combinatorialoptimization problem onto a model Hamiltonian H_(C)=Σ_(α=1)^(t)h_(α)P_(α), a set of variational parameters ({right arrow over(θ)}=θ₁, θ₂, . . . , θ_(N) for the VQE algorithm, ({right arrow over(γ)}=γ₁, γ₂, . . . , γ_(p), {right arrow over (β)}=β₁, β₂, . . . ,β_(p)) for the QAOA) is selected, by the classical computer 102, toconstruct a sequence of gates (also referred to a “trial statepreparation circuit”) A({right arrow over (θ)}) for the VQE or A({rightarrow over (γ)}, {right arrow over (β)}) for the QAOA, which preparesthe quantum processor 106 in a trial state |Ψ({right arrow over (θ)})

for the VQE or |Ψ({right arrow over (γ)}, {right arrow over (β)})

for the QAOA. For the initial iteration, a set of variational parameters{right arrow over (θ)} in the VQE may be chosen randomly. In the QAOA, aset of variational parameters {right arrow over ((γ)}, {right arrow over(β)}) may be randomly chosen for the initial iteration.

This trial state |Ψ({right arrow over (θ)})

, |Ψ({right arrow over (γ)}, {right arrow over (β)})

is used to provide an expectation value of the model Hamiltonian H_(C).

In the VQE algorithm, the trial state preparation circuit A({right arrowover (θ)}) may be constructed by known methods, such as the unitarycoupled cluster method, based on the model Hamiltonian H_(C) and theselected set of variational parameters {right arrow over (θ)}.

In the QAOA, the trial state preparation circuit A({right arrow over(γ)}, {right arrow over (β)}) includes p layers (i.e., p-timerepetitions) of an entangling circuit U(γ_(l)) that relates to the modelHamiltonian H_(C)(U(γ_(l))=e^(−iγ) ^(l) ^(H) ^(C) ) and a mixing circuitU_(Mix)(β_(l)) that relates to a mixing term H_(B)=Σ_(i=1) ^(n)σ_(i)^(X)(U_(Mix)(β_(l))=e^(−β) ^(l) ^(H) ^(B) ) (l=1, 2, . . . , p) asA({right arrow over (γ)},{right arrow over (β)})=U _(Mix)(β_(p))U_(Mix)(β_(p-1))U(γ_(p-1)) . . . U _(Mix)(β₁)U(γ₁).

Each term σ_(i) ^(X) (in the mixing term H_(B) corresponds to aπ/2-pulse (as described above in relation to FIG. 4 ) applied to qubit iin the quantum processor 106.

To allow the application of the trial state preparation circuit A({rightarrow over (θ)}), A({right arrow over (γ)}, {right arrow over (β)}) on aNISQ device, the number of the quantum gate operations need to be small(i.e., shallow circuits) such that errors due to the noise in the NISQdevice are not accumulated. However, as the problem size increases, thecomplexity of the trial state preparation circuit A({right arrow over(θ)}), A({right arrow over (γ)}, {right arrow over (β)}) may increaserapidly, leading to deep circuits (i.e., an increased number of timesteps required to execute gate operations in circuits to construct)required to construct the trial state preparation circuit A({right arrowover (θ)}), A({right arrow over (γ)}, {right arrow over (β)}).Furthermore, some trial state preparation circuit A({right arrow over(θ)}), A({right arrow over (γ)}, {right arrow over (β)}) that aredesigned hardware-efficiently with shallow circuits (i.e., a decreasednumber of time steps required to execute gate operations) may notprovide a large enough variational search space to find the lowestenergy of the model Hamiltonian H_(C). In the embodiments describedherein, the terms in the model Hamiltonian H_(C) are grouped intosub-Hamiltonians H_(λ)(λ=1, 2, . . . , u), where u is the number ofsub-Hamiltonians (i.e., H_(C)=Σ_(λ=1) ^(u)H_(λ)), and the trial statepreparation circuit A({right arrow over (θ)}), A({right arrow over (γ)},{right arrow over (β)}) is replaced with a reduced state preparationcircuit A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC) ^(λ)({right arrowover (γ)}, {right arrow over (β)}) to evaluate an expectation value ofeach sub-Hamiltonian H_(λ). The reduced state preparation circuitA_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC) ^(λ)({right arrow over(γ)}, {right arrow over (β)}) for a sub-Hamiltonian H_(λ) is constructedby a set of gate operations that can influence an expectation value ofthe sub-Hamiltonian H_(λ) (referred to as the past causal cone (PCC) ofthe sub-Hamiltonian). Other gate operations (that do not influence theexpectation value of the sub-Hamiltonian H_(λ)) in the trial statepreparation circuit A({right arrow over (θ)}), A({right arrow over (γ)},{right arrow over (β)}) are removed in the reduced state preparationcircuits A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC) ^(λ)({right arrowover (γ)}, {right arrow over (β)}). In some embodiments,sub-Hamiltonians H_(λ)of the model Hamiltonian H_(C) may respectivelycorrespond to Pauli terms P_(α) in the model Hamiltonian H_(C). In someembodiments, a sub-Hamiltonian H_(λ) is a collection of more than onePauli terms P_(α) in the model Hamiltonian H_(C).

For example, the model Hamiltonian H_(C)=σ₁ ^(Z)⊗σ₂ ^(Z)+σ₂ ^(Z)⊗σ₃^(Z)+σ₃ ^(Z)⊗σ₄ ^(Z)+σ₁ ^(Z)⊗σ₄ ^(Z) may be grouped into foursub-Hamiltonians, H₁=σ₁ ^(Z)⊗σ₂ ^(Z), H₂=σ₂ ^(Z)⊗σ₃ ^(Z), H₃=σ₃ ^(Z)⊗σ₄^(Z), and H₄=σ₁ ^(Z)⊗σ₄ ^(Z). This model Hamiltonian may be defined on asystem of five superconducting qubits (qubit 1, 2, . . . , 5) aligned ina one dimensional array, in which two-qubit gates can be applied betweenneighboring pairs of qubits (1, 2), (2, 3), (3, 4), (4, 5). FIG. 7Aillustrates the trial state preparation circuit (A({right arrow over(γ)}, {right arrow over (β)})=U (γ₁)U_(Mix)(β₁)) 700, where p=1. Themixing circuit U_(Mix)(β_(l)) can be implemented by single-qubitrotation gates 702, 704, 706, 708 on qubits 1, 2, 3, and 4,respectively. The entangling circuit U(γ₁) is related to the modelHamiltonian H_(C) as described above. The first term σ₁ ^(Z) ⊗σ₂ ^(Z) inthe model Hamiltonian H_(C) can be implemented in combination ofcontrolled-NOT gates 710 on qubit 2 conditioned on qubit 1 and targetedon qubit 2, and a single-qubit rotation gate 712 on qubit 2 about theZ-axis of the Bloch sphere 400 by a polar angle γ₁/2. As one willappreciate, the implementation of such gates can be performed bycombining properly adjusted entangling interaction gate operationbetween qubits 1 and 2 and composite pulses applied to qubits 1 and 2.The terms σ₂ ^(Z)⊗σ₃ ^(Z) and σ₃ ^(Z)⊗σ₄ ^(Z) in the model HamiltonianH_(C) can be implemented similarly. The last term σ₁ ^(Z)⊗σ₄ ^(Z) in themodel Hamiltonian H_(C) can be implemented in combination ofcontrolled-NOT gates conditioned on qubit 1 and targeted on qubit 2,controlled-NOT gates conditioned on qubit 2 and targeted on qubit 3,controlled-NOT gates conditioned on qubit 3 and targeted on qubit 4, andcontrolled-NOT gates conditioned on qubit 4 and targeted on qubit 5 andsingle-qubit rotation gates on qubits 1, 2, 3, 4, and 5. FIG. 7Billustrates the reduced state preparation circuits (A_(PCC) ¹({rightarrow over (γ)}, {right arrow over (β)})) 714 to evaluate an expectationvalue of the sub-Hamiltonian H₁=σ₁ ^(Z)⊗σ₂ ^(Z). Since qubits 3 and 4 donot affect the expectation value of the sub-Hamiltonian H₁=σ₁ ^(Z)⊗σ₂^(Z), the controlled-NOT gates and the single-qubit rotation gates thatare applied only to qubits 3 and 4 in the trial state preparationcircuit A({right arrow over (γ)}, {right arrow over (β)}) are removed.The reduced state preparation circuits A_(PCC) ^(λ)({right arrow over(γ)}, {right arrow over (β)}) to evaluate an expectation value of othersub-Hamiltonians H_(λ) can be constructed similarly.

With the reduced trial state preparation circuit A_(PCC) ^(λ)({rightarrow over (θ)}), A_(PCC) ^(λ)({right arrow over (γ)}, {right arrow over(β)}) for a sub-Hamiltonian H_(λ), a trial state |Ψ_(λ)({right arrowover (θ)})

, Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

is prepared on the quantum processor 106 to evaluate an expectation ofthe sub-Hamiltonian H_(λ). This step is repeated for all of thesub-Hamiltonians H_(λ) (λ=1, 2, . . . , u). The expectation value of themodel Hamiltonian H_(C) is a sum of the expectation values of all of thesub-Hamiltonians H_(λ)(λ=1, 2, . . . , u). The use of the reduced trialstate preparation circuit A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC)^(λ)({right arrow over (γ)}, {right arrow over (β)}) reduces the numberof gate operations to apply on the quantum processor 106. Thus, a trialstate |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

can be constructed without accumulating errors due to the noise in theNISQ device.

In block 606, following the selection of a set of variational parameters{right arrow over (θ)}, ({right arrow over (γ)}, {right arrow over(β))}, the quantum processor 106 is set in an initial state |Ψ₀

by the system controller 104. In the VQE algorithm, the initial state|Ψ₀

may correspond to an approximate ground state of the system that iscalculated by a classical computer or an approximate ground state thatis empirically known to one in the art. In the QAOA algorithm, theinitial state |Ψ₀

may be in the hyperfine ground state of the quantum processor 106 (inwhich all qubits are in the uniform superposition over computationalbasis states (in which all qubits are in the superposition of |0

and |1

, |0

+|1

). A qubit can be set in the hyperfine ground state |0

by optical pumping and in the superposition state |0

+|1

by application of a proper combination of single-qubit operations(denoted by “H” in FIG. 7 ) to the hyperfine ground state |0

.

In block 608, following the preparation of the quantum processor 106 inthe initial state |Ψ₀

, the trial state preparation circuit A({right arrow over (θ)}),A({right arrow over (γ)}, {right arrow over (β)}) is applied to thequantum processor 106, by the system controller 104, to construct thetrial state |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

for evaluating an expectation of the sub-Hamiltonian H_(λ). The reducedtrial state preparation circuit A_(PCC) ^(λ)({right arrow over (θ)}),A_(PCC) ^(λ)({right arrow over (γ)}, {right arrow over (β)}) isdecomposed into series of two-qubit entanglement gate operations(entangling gates) and single-qubit operations (R gates) and optimizedby the classical computer 102. The series of two-qubit entanglement gateoperations (entangling gates) and single-qubit operations (R gates) canbe implemented by application of a series of pulses, intensities,durations, and detuning of which are appropriately adjusted by theclassical computer 102 on the set initial state |Ψ₀

and transform the quantum processor from the initial state |Ψ₀

to trial state |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

.

In block 610, following the construction of the trial state|Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

on the quantum processor 106, the expectation value F_(λ)({right arrowover (θ)})=

Ψ_(λ)({right arrow over (θ)})|H_(λ)|Ψ_(λ)({right arrow over (θ)})

, F_(λ)({right arrow over (γ)}, {right arrow over (β)})=

Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})|H_(λ)|Ψ_(λ)({rightarrow over (γ)}, {right arrow over (β)})

of the sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u) is measured by thesystem controller 104. Repeated measurements of states of thesuperconducting qubits in the quantum processor 106 in the trial state|Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

yield the expectation value the sub-Hamiltonian H_(λ).

In block 612, following the measurement of the expectation value of thesub-Hamiltonian H_(λ)(λ=1, 2, . . . , u), blocks 606 to 610 for anothersub-Hamiltonian H_(λ)(λ=1, 2, . . . , u) until the expectation values ofall the sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u) in the modelHamiltonian H_(C)=Σ_(λ=1) ^(u)H_(λ) have been measured by the systemcontroller 104.

In block 614, following the measurement of the expectation values of allthe sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u), a sum of the measuredexpectation values of all the sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u)of the model Hamiltonian H_(C)=Σ_(λ=1) ^(u)H_(λ) (that is, the measuredexpectation value of the model Hamiltonian H_(C), F({right arrow over(θ)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (θ)}), F({right arrow over(γ)}, {right arrow over (β)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (γ)},{right arrow over (β)}) is computed, by the classical computer 102.

In block 616, following the computation of the measured expectationvalue of the model Hamiltonian H_(C), the measured expectation valueF({right arrow over (γ)}, {right arrow over (β)}) of the modelHamiltonian H_(C) is compared to the measured expectation value of themodel Hamiltonian H_(C) in the previous iteration, by the classicalcomputer 102. If a difference between the two values is less than apredetermined value (i.e., the expectation value sufficiently convergestowards a fixed value), the method proceeds to block 620. If thedifference between the two values is more than the predetermined value,the method proceeds to block 618.

In block 618, another set of variational parameters {right arrow over(θ)}, ({right arrow over (γ)}, {right arrow over (β))} for a nextiteration of blocks 606 to 616 is computed by the classical computer102, in search for an optimal set of variational parameters {right arrowover (θ)}, {right arrow over ((γ)}, {right arrow over (β))} to minimizethe expectation value of the model Hamiltonian H_(C), F({right arrowover (θ)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (θ)}), F({right arrowover (γ)}, {right arrow over (β)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over(γ)}, {right arrow over (β)}). That is, the classical computer 102 willexecute a classical optimization method to find the optimal set ofvariational parameters

$\overset{\rightarrow}{\theta},{\left( {\overset{\rightarrow}{\gamma},\overset{\rightarrow}{\beta}} \right){\left( {{\min\limits_{\overset{\rightarrow}{\theta}}{F\underset{\overset{\rightarrow}{\gamma},\overset{\rightarrow}{\beta}}{\left( \overset{\rightarrow}{\theta} \right)}}},{\min\;{F\left( {\overset{\rightarrow}{\gamma},\overset{\rightarrow}{\beta}} \right)}}} \right).}}$Example of conventional classical optimization methods includesimultaneous perturbation stochastic approximation (SPSA), particleswarm optimization (PSO), Bayesian optimization (BO), and Nelder-Mead(NM).

In block 620, the classical computer 102 will typically output theresults of the variational search to a user interface of the classicalcomputer 102 and/or save the results of the variational search in thememory of the classical computer 102. The results of the variationalsearch will include the measured expectation value of the modelHamiltonian H_(C) in the final iteration corresponding to the minimizedenergy of the system in the selected quantum chemistry problem, or theminimized value of the objective function C(z)=Σ_(α=1) ^(t)h_(α)C_(α)(z)of the selected combinatorial optimization problem (e.g., a shortestdistance for all of the trips visiting all given cities in a travellingsalesman problem) and the measurement of the trail state |Ψ_(λ)({rightarrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

in the final iteration corresponding to the configuration of electronsor spins that provides the lowest energy of the system, or the solutionto the N-bit string (z=z₁z₂ . . . z_(N)) that provides the minimizedvalue of the objective function C(z)=E_(α=1) ^(t)h_(α)C_(α) (z) of theselected combinatorial optimization problem (e.g., a route of the tripsto visit all of the given cities that provides the shortest distance fora travelling salesman).

The variational search reduced trial state preparation circuitsdescribed herein provides an improved method for obtaining a solution toan optimization problem by the Variational Quantum Eigensolver (VQE)algorithm or the Quantum Approximate Optimization Algorithm (QAOA) on ahybrid quantum-classical computing system. Thus, the feasibility that ahybrid quantum-classical computing system may allow solving problems,which are not practically feasible on classical computers, or suggest aconsiderable speed up with respect to the best known classical algorithmeven with a noisy intermediate-scale quantum device (NISQ) device.

While the foregoing is directed to specific embodiments, other andfurther embodiments may be devised without departing from the basicscope thereof, and the scope thereof is determined by the claims thatfollow.

The invention claimed is:
 1. A hybrid quantum-classical computingsystem, comprising: a quantum processor comprising a group of Josephsonjunctions, each of the Josephson junctions having two energy separatedstates defining a superconducting qubit; one or more microwave waveformgenerator configured to irradiate, which is provided to the Josephsonjunctions in the quantum processor; and a classical computer, whereinthe hybrid quantum-classical computing system is configured to: compute,by the classical computer, a model Hamiltonian onto which a selectedproblem is mapped, wherein the model Hamiltonian comprises a pluralityof sub-Hamiltonians; execute iterations, each iteration comprising:setting the quantum processor in an initial state; transform the quantumprocessor from the initial state to a trial state based on each of theplurality of sub-Hamiltonians and an initial set of variationalparameters by applying a first trial state preparation circuit to thequantum processor; measuring an first value of each of the plurality ofsub-Hamiltonians on the quantum processor; and selecting another set ofvariational parameters based on a classical optimization method if it isdetermined that a difference between the measured expectation values ofthe model Hamiltonian in the current iteration and the previousiteration is more than a predetermined value and then: outputs themeasured expectation value of the model Hamiltonian as an optimizedsolution to the selected problem if it is determined that the differenceis less than the predetermined value.
 2. The hybrid quantum-classicalcomputing system according to claim 1, wherein the first trial statepreparation circuit does not include gate operations that do notinfluence the expectation value of the each of the plurality ofsub-Hamiltonians.
 3. The hybrid quantum-classical computing systemaccording to claim 1, wherein if it is determined that the difference ismore than the predetermined value, the classical computer executesanother iteration.
 4. The hybrid quantum-classical computing systemaccording to claim 1, wherein the selected problem to be solved isfinding a lowest energy of a many-particle quantum system.
 5. The hybridquantum-classical computing system according to claim 4, wherein theclassical computer further selects the initial set of variationalparameters randomly.
 6. The hybrid quantum-classical computing systemaccording to claim 4, wherein the initial state is an approximate stateof the many-particle quantum system that is calculated by the classicalcomputer.
 7. The hybrid quantum-classical computing system according toclaim 1, wherein the selected problem to be solved is a combinatorialoptimization problem.
 8. The hybrid quantum-classical computing systemaccording to claim 7, wherein the classical computer selects the initialset of variational parameters randomly.
 9. The hybrid quantum-classicalcomputing system according to claim 7, wherein the initial state is asuperposition of the two energy separated states.
 10. A hybridquantum-classical computing system comprising non-volatile memory havinga number of instructions stored therein which, when executed by one ormore processors, causes the hybrid quantum-classical computing system toperform operations comprising: computing, by a classical computer, amodel Hamiltonian onto which a selected problem is mapped, wherein themodel Hamiltonian comprises a plurality of sub-Hamiltonians; executingiterations, each iteration comprising: setting a quantum processor in aninitial state, wherein the quantum processor comprises a plurality ofJosephson junctions, each of which has two frequency-separated statesdefining a qubit; transforming the quantum processor from the initialstate to a trial state based on each of the plurality ofsub-Hamiltonians and an initial set of variational parameters byapplying a first trial state preparation circuit to the quantumprocessor; measuring an expectation value of each of the plurality ofsub-Hamiltonians on the quantum processor; and selecting another set ofvariational parameters based on a classical optimization method if it isdetermined that a difference between the measured expectation values ofthe model Hamiltonian in the current iteration and the previousiteration is more than a predetermined value; and outputting themeasured expectation value of the model Hamiltonian as an optimizedsolution to the selected problem if it is determined that the differenceis less than the predetermined value.
 11. The hybrid quantum-classicalcomputing system according to claim 10, wherein the first trial statepreparation circuit does not include gate operations that do notinfluence the expectation value of the each of the plurality ofsub-Hamiltonians.